Chapter 1 · Algebraic Proofs · Concept 2
1-4 Division of Polynomials
Think of a food delivery app. The number of orders each day grows faster and faster, but new riders are hired at a steady rate. To find how many orders one rider has to cover, you divide the big order function by the rider function.
Guiding question
How does dividing algebraic expressions help us find the share of a single unit inside a system that keeps growing?
Learning outcome. Performs polynomial division using standard algorithm.
You have done long division since you were small. Dividing polynomials is the same thing, with x in it.
Numbers
7 ÷ 3 → 2 left over 1
You take away as many 3s as you can. What will not fit is the remainder.
Polynomials
(x² − 5x + 1) ÷ (x − 2) → x − 3 left over −5
Same idea. You stop when what is left has a smaller power than what you are dividing by.
Division of polynomials can be worked out using long division, just like whole numbers.
In problems that say “treat as a polynomial in x”, treat every other letter as a number.
So in 4x2 − 9ax − 9a2, the letter a is just a number that has not been told to you. Only the powers of x decide what goes where.
A finished division is a page of numbers with no arrows on it. Here it is built one line at a time, and each line says what just happened. Type your own, or tap one of the book's questions.
Type powers however you like — 2x^3, 2x3 and 2x³ all work.
1. Find the quotient and the remainder when 2x3 − 5x2 − 2 is divided by 2x2 − x − 1.
- 1
Notice that a power is missing.
2x3 − 5x2 + 0x − 2There is no x term. Leave a gap for it, or your columns will slide across and every line after will be wrong.
2. For A = 4x2 − 9ax − 9a2 and B = x − 3a, treating each as a polynomial in x, find the quotient and the remainder when A is divided by B.
- 1
Decide what counts as a number here.
a is a number. Only x is the letter we divide by.That is what "as a polynomial in x" means. So −9ax is a number times x.
Work with a partner. Solve the Try section with your group, then talk about the answer. For question 2, treat it as a polynomial in x.
Find the quotient and the remainder when A is divided by B. For 7–10, treat as a polynomial in x.
A box shaped like a cuboid has volume 4x3 + 4x2 − 9x − 9. The area of its base is 2x2 − x − 3. Write its height in terms of x.
Multiply by the WHOLE divider.
Dividing by 2x² − x − 1 and then multiplying only by the 2x² is the commonest slip on this page. Every term of the divider gets multiplied, every time.
Taking away a negative adds.
(−5x²) − (−x²) is −4x², not −6x². Most lost marks in a long division are one subtraction, not the method. Write the signs down before you work them out.
Leave a gap for a missing power.
In x³ − 4x + 1 there is no x² term. Leave that column empty. If you close the gap, every line below slides across and the whole answer is wrong.
Know when to stop.
Stop when what is left has a SMALLER highest power than the divider. Not when it looks tidy, and not only when it reaches zero — the remainder is often not zero.
Keep reading — this lesson marks itself done automatically.